Generally speaking, the relationship between two variables X and Y can be expressed as a function with the shape of y = kx (where K is a constant and k≠0), then Y is called the proportional function of X..

Proportional function belongs to linear function, but linear function is not necessarily proportional function. Proportional function is a special form of linear function, that is, in the linear function y=kx+b, if b = 0, that is, the so-called "Y-axis intercept" is zero, it is a proportional function. The relationship of the proportional function is expressed as: y=kx(k is the proportional coefficient).

When k > 0 (one or three quadrants), the larger k is, the closer the image is to the Y axis. The function value y increases with the increase of independent variable X.

When k < 0 (24 quadrants), the smaller k is, the closer the image is to the Y axis. When the value of independent variable x increases, the value of y decreases gradually.

[Edit this paragraph] Properties of proportional function

1. domain: r (real number set)

2. Scope: R (real number set) ... extends the concept of full-text proportional function.

Generally speaking, the relationship between two variables X and Y can be expressed as a function with the shape of y = kx (where K is a constant and k≠0), then Y is called the proportional function of X..

Proportional function belongs to linear function, but linear function is not necessarily proportional function. Proportional function is a special form of linear function, that is, in the linear function y=kx+b, if b = 0, that is, the so-called "Y-axis intercept" is zero, it is a proportional function. The relationship of the proportional function is expressed as: y=kx(k is the proportional coefficient).

When k > 0 (one or three quadrants), the larger k is, the closer the image is to the Y axis. The function value y increases with the increase of independent variable X.

When k < 0 (24 quadrants), the smaller k is, the closer the image is to the Y axis. When the value of independent variable x increases, the value of y decreases gradually.

[Edit this paragraph] Properties of proportional function

1. domain: r (real number set)

2. Range: R (real number set)

3. Parity check: odd function

4. Monotonicity: When k>0, the image is located in the first and third quadrants, and Y increases (monotonically) with the increase of X; When k < 0, the image is located in the second and fourth quadrants, and y decreases (monotonically decreases) with the increase of x.

5. Periodicity: Not a periodic function.

6. Symmetry axis: straight line, no symmetry axis.

[Edit this paragraph] Solution of proportional resolution function

Let the analytic formula of the proportional function be y=kx(k≠0), and bring the coordinates of known points into the above formula to get k, then the analytic formula of the proportional function can be obtained.

In addition, if you want to find the intersection coordinates of proportional function and other functions, you can combine two known resolving function equations to find their x and y values.

[Edit this paragraph] Image of proportional function

The image of the proportional function is a straight line passing through the coordinate origin (0,0) and the fixed point (x, kx). Its slope is k, and the horizontal and vertical intercepts are 0.

[Edit this paragraph] Practice of the image of proportional function

1. Take a value within the allowable range of x, and calculate the value of y according to the analytical formula.

2. Draw a point according to the values of x and y obtained in the first step.

3. The straight line between the point drawn in the second step and the origin.

[Edit this paragraph] Application of proportional function

The power of positive proportional function in linear programming problem is infinite.

For example, the slope problem depends on the value of k, and the greater the k, the greater the angle between the function image and the X axis, and vice versa.

Also, y=kx is the symmetry axis of the image with y = k/x.

① Proportion: two related quantities, one of which changes and the other changes accordingly. If the ratio (that is, quotient) of the two numbers corresponding to these two quantities is certain, these two quantities are called proportional quantities, and the relationship between them is called proportional relationship. ① Represented by letters: If the letters X and Y are used to represent two related quantities and K is used to represent their ratio, the (certain) proportional relationship can be used as follows.

(2) the changing law of two related quantities in direct proportion: for direct proportion, y = kx(k & gt；; 0), at this time, y and x expand and contract at the same time, and the ratio remains unchanged. For example, the speed of a car per hour is constant, and the distance traveled is directly proportional to the time spent?

The above manufacturers are certain, so dividend and divisor represent two related quantities, which are in direct proportion. Note: When judging whether two related quantities are directly proportional, we should pay attention to these two related quantities. Although they are also a quantity, they change with the change of another, but the proportion of the two numbers they correspond to is not necessarily, so they cannot be directly proportional. Such as a person's age and weight.

[Edit this paragraph] Definition of inverse proportional function

Generally speaking, if the relationship between two variables X and Y can be expressed by Y = K/X (where K is a constant and k≠0), then Y is said to be an inverse proportional function of X. ..

Because y=k/x is a fraction, the range of the independent variable x is X≠0. And y=k/x is sometimes written as xy=k or y=kx- 1.

[Edit this paragraph] Inverse proportional function expression

Y = k/x where x is an independent variable and y is a function of x.

y=k/x=k 1/x

xy=k

y=k x^- 1

Y=k\x(k is a constant (k≠0, x is not equal to 0).

[Edit this paragraph] Independent variable value range of inverse proportional function

①k≠0； (2) In general, the range of the independent variable X is a real number of x ≠ 0; (3) The range of function y is also all non-zero real numbers.

[Edit this paragraph] Inverse proportional function image

The image of inverse proportional function belongs to hyperbola,

The curve is getting closer to the X axis and the Y axis, but it will not intersect (K≠0).

[Edit this paragraph] Properties of inverse proportional function

1. When k>0, the image is located in the first and third quadrants respectively; When k < 0, the image is located in the second and fourth quadrants respectively.

2. When k>0 is in the same quadrant, Y decreases with the increase of X; When k < 0, y increases with the increase of x in the same quadrant.

K>0, the functions are all subtraction functions on x0; K<0, these two functions are increasing function on x0.

The domain is x ≠ 0; The range is y≠0.

3. Because in y=k/x(k≠0), X can't be 0 and Y can't be 0, so the image of inverse proportional function can't intersect with X axis or Y axis.

4. In the inverse proportional function image, take any two points P and Q, the intersection points P and Q are parallel lines of the X axis and the Y axis respectively, the rectangular area enclosed with the coordinate axis is S 1, S2 is S 1 = S2 = | k |.

5. The image of inverse proportional function is not only an axisymmetric figure, but also a centrally symmetric figure. It has two symmetry axes y=x y=-x (that is, the bisectors of the first, third and fourth quadrants), and the center of symmetry is the coordinate origin.

6. If the positive proportional function y=mx and the inverse proportional function y=n/x intersect at two points A and B (m the signs of m and n are the same), then the two points A and B are symmetrical about the origin.

7. Let there be an inverse proportional function y=k/x and a linear function y=mx+n on the plane. If they have a common intersection, B2+4k m ≥ (not less than) 0.

8. Inverse proportional function y = k/x: asymptote of X axis and Y axis.

[Edit this paragraph] Application example of inverse proportional function

For example, there is a point P(m, n) on the image of 1 inverse proportional function, whose coordinates are two of the unary quadratic equation t2-3t+k=0 about t, and the distance from P to the origin is the root sign 13. Find the analytic expression of inverse proportional function.

Analysis:

To find the inverse proportional resolution function is to find k, so we need to list an equation about k.

Solution: ∫m, n is two of the equation t2-3t+k=0 about t.

∴ m+n=3，mn=k，

PO= root number 13,

∴ m2+n2= 13，

∴(m+n)2-2mn= 13，

∴ 9-2k= 13。

∴ k=-2

When k=-2 and delta = 9+8 > 0,

∴ k=-2 meets the requirements,

Example 2 The straight line intersects the hyperbola located in the second quadrant at two points A and A 1. After passing point A, it is perpendicular to the X axis and Y axis, with vertical feet of B and C respectively, and the area of right-angled ABOC is 6. Find:

(1) Analytical expressions of straight lines and hyperbolas;

(2) The coordinates of point A and A 1 point.

Analysis: AB side and AC side of rectangular ABOC are vertical line segments from point A to X axis and Y axis respectively.

Let the coordinates of point A be (m, n), then AB=|n|, AC=|m|,

According to the area formula of rectangle | m n | = 6.

Example 3: As shown in the figure, there are two points A and C perpendicular to the X axis, and the vertical feet are B and D respectively, connecting OC and OA. Let OC and AB intersect at E, the area of △AOE is S 1, and the area of quadrilateral BDCE is S2. Compare the size of S 1 and S2.

[Edit this paragraph] Mathematical terminology

Pronunciation y

Explain the basic concept of function: Generally speaking, in a change process, there are two variables X and Y, and for each definite value of X, Y has a unique definite value corresponding to it. Then we say that X is an independent variable and Y is a function of X, which is expressed as y = kx+b (where B is an arbitrary constant and K is not equal to 0). When b = 0, y is the proportional function of x, and the proportional function is a special case of linear function. It can be expressed as y=kx.

[Edit this paragraph] Basic definition

Variable: the number of changes

Constant: a constant quantity

The independent variable x and the linear function y of x have the following relationship:

Y=kx+b (k is an arbitrary non-zero constant and b is an arbitrary constant)

When x takes a value, y has one and only one value corresponding to x, and if there are two or more values corresponding to x, it is not a linear function.

X is an independent variable, y is a dependent variable, k is a constant, and y is a linear function of X.

Especially, when b=0, y is the proportional function of x, that is, the image of the proportional function of y=kx (k is constant, but K≠0) passes through the origin.

Domain: the range of independent variables should make the function meaningful; It should be realistic.

[Edit this paragraph] Related properties

Functional attribute

The change value of 1.y is directly proportional to the corresponding change value of x, and the ratio is k.

That is: y=kx+b(k≠0) (k is not equal to 0, and k and b are constants).

2. When x=0, b is a function on the Y axis, and the coordinate is (0, b).

3.k is the slope of the linear function y=kx+b, and k = tan θ (the angle θ is the included angle between the linear function image and the positive direction of the X axis, θ ≠ 90).

Form, take, image, intersection and subtraction.

4. When b=0 (y=kx), the image of a linear function becomes a proportional function, which is a special linear function.

5. Function image properties: when k is the same and b is not equal, the images are parallel; When k is different and b is equal, the images intersect; When k is negative reciprocal, two straight lines are vertical; When k and b are the same, the two straight lines coincide.

Image attribute

1. Practice and graphics: Through the following three steps.

(1) list

(2) tracking points; [Generally, two points are taken and a straight line is determined by two points];

(3) The connection can be the image of a function-a straight line. So the image of a function only needs to know two points and connect them into a straight line. (Usually, the intersections of the function image with the X axis and the Y axis are -k points B and 0, 0 and B, respectively. )

2. Property: any point P(x, y) on the (1) linear function satisfies the equation: y=kx+b(k≠0). (2) The coordinates of the linear function intersecting with the Y axis are always (0, b), and the images of the proportional function intersecting with the X axis at (-b/k, 0) are all at the origin.

3. Function is not a number, it refers to the relationship between two variables in a certain change process.

4. Quadrant where K, B and function images are located:

When y=kx (that is, b is equal to 0 and y is proportional to x):

When k > 0, the straight line must pass through the first and third quadrants, and y increases with the increase of x;

When k < 0, the straight line must pass through the second and fourth quadrants, and y decreases with the increase of x.

When y=kx+b:

When k>0, b>0, then the image of this function passes through the first, second and third quadrants.

When k>0, b<0, then the image of this function passes through the first, third and fourth quadrants.

When k0, the image of this function passes through the first, second and fourth quadrants.

When k < 0, b<0, then the image of this function passes through the second, third and fourth quadrants.

When b > 0, the straight line must pass through the first and second quadrants;

When b < 0, the straight line must pass through the third and fourth quadrants.

Particularly, when b=0, the image of the proportional function is represented by a straight line of the origin o (0 0,0).

At this time, when k > 0, the straight line only passes through the first and third quadrants, but not the second and fourth quadrants. When k < 0, the straight line only passes through the second and fourth quadrants, but not through the first and third quadrants.

4. Special positional relationship

When two straight lines in the plane rectangular coordinate system are parallel, the k value in the resolution function (that is, the coefficient of the first term) is equal.

When two straight lines are perpendicular to each other in the plane rectangular coordinate system, the value of k in the resolution function is negative reciprocal (that is, the product of two values of k is-1).

[Edit this paragraph] Expression

Analytical type

①ax+by+c=0 [general formula]

②y=kx+b[ oblique]

(k is the slope of the straight line, b is the longitudinal intercept of the straight line, and the proportional function b=0).

③y-y 1=k(x-x 1)[ point inclination]

(k is the slope of the straight line, (x 1, y 1) is the point where the straight line passes)

④ (y-y1)/(y2-y1) = (x-x1)/(x2-x1) [two-point formula]

((x 1, y 1) and (x2, y2) are two points on a straight line)

⑤x/a-y/b=0[ intercept type]

(A and B are the intercepts of a straight line on the X axis and the Y axis, respectively)

Limitations of analytical expressions:

① More requirements (3);

② and ③ cannot express straight lines without slope (straight lines parallel to the X axis);

④ There are many parameters and the calculation is too complicated;

⑤ Cannot represent a straight line parallel to the coordinate axis and a straight line passing through a point.

Inclination angle: The included angle between the X axis and the straight line (the angle formed by the straight line and the positive direction of the X axis) is called the inclination angle of the straight line. Let the inclination of the straight line be a, and the slope of the straight line be k=tg(a).

[Edit this paragraph] Common formulas

1. Find the k value of the function image: (y 1-y2)/(x 1-x2).

2. Find the midpoint of the line segment parallel to the X axis: |x 1-x2|/2.

3. Find the midpoint of the line segment parallel to the Y axis: |y 1-y2|/2.

4. Find the length of any line segment: √ (x 1-x2) 2+(y 1-y2) 2 (note: the sum of squares of (x1-x2) and (y1-y2) under the root sign).

5. Use a linear function to find the intersection coordinates of two images: solve two functions.

Two linear functions y 1 = k1x+y1= y2 = k2x+B2 make y 1x+b 1 = k2x+b2 replace the solution value of x=x0 back to y1=

6. Find the midpoint coordinates of a line segment connected by any two points: [(x 1+x2)/2, (y 1+y2)/2].

7. Find the first resolution function of any two points: (x-x1)/(x1-x2) = (y-y1)/(y1-y2) (where the denominator is 0 and the numerator is 0).

x y

++in the first quadrant

+-In the fourth quadrant

-+in the second quadrant

-In the third quadrant

8. If two straight lines y1= k1x+b1‖ y2 = k2x+b2, then k 1=k2, b 1≠b2.

9. If two straight lines y1= k1x+b1⊥ y2 = K2x+B2, then k 1×k2=- 1.

10.

Y=k(x-n)+b is to translate n units to the right.

Y=k(x+n)+b is to translate n units to the left.

Formula: right minus left plus (for y=kx+b, only change K)

Y=kx+b+n is to translate up by n units.

Y=kx+b-n is a downward translation of n units.

Formula: increase or decrease (for y=kx+b, only change b)

[Edit this paragraph] Related applications

Application in life

1. When the time t is constant, the distance s is a linear function of the velocity v .. s=vt.

2. When the pumping speed f of the pool is constant, the water quantity g in the pool is a linear function of the pumping time t .. Set the original water quantity in the pool. G = S- feet.

3. When the original length b of the spring (the length when the weight is not hung) is constant, the length y of the spring after the weight is hung is a linear function of the weight x, that is, y=kx+b(k is an arbitrary positive number).

mathematical problem

First, determine the range of the letter coefficient.

Example 1 If the proportional function is known, then when k

Solution: According to the definition and properties of proportional function, M is obtained.

Second, compare the size of x value or y value.

Example 2. Given that points P 1(x 1, y 1) and P2(x2, y2) are two points on the image of linear function y=3x+4, Y 1 >: Y2, then the relationship between x 1 and x2 is ().

A.x 1 & gt； x2 b . x 1 & lt； X2c.x 1 = X2D。 Can't be sure.

Solution: according to the meaning of the question, k = 3>0 and y1>; Y2. according to the property of linear function "when k>0, y increases with the increase of x", x1>; X2. So choose A..

Thirdly, judge the position of the function image.

Example 3. The linear function y=kx+b satisfies kb >；; 0, and y decreases with the increase of x, then the image of this function does not pass ().

A. The first quadrant B. The second quadrant

C. The third quadrant D. The fourth quadrant

Solution: Through kb>0, we know that K and B have the same number. Because y decreases with the increase of x, k

Typical example

Example 1. A spring, without hanging object 12cm, will extend after hanging the object, and the length of extension is proportional to the mass of the suspended object. If the total length of the spring is 13.5cm after a 3kg object is suspended, find the functional relationship between the total length of the spring and the mass x(kg) of the suspended object. If the maximum total length of the spring is

Analysis: This problem has changed from a qualitative problem in physics to a quantitative problem in mathematics, which is also a practical problem. Its core is that the total length of the spring is the sum of the unloaded length and the loaded extension length, and the range of independent variables can be handled by the maximum total length → maximum extension → maximum mass and practical thinking.

Solution: Set the function as y=kx+ 12 from the meaning of the question.

Then 13.5=3k+ 12, and k=0.5.

The resolution function is y=0.5x+ 12.

From 23=0.5x+ 12: x=22。

The value range of the independent variable x is 0≤x≤22.

A school needs to burn some computer CDs. If you burn in a computer company, you need 8 yuan for each CD. If you burn it yourself, you need 4 yuan for each CD, in addition to renting the burner of 120 yuan. Do you want to burn these CDs in the computer company or burn them yourself?

This question should consider the range of X.

Solution: let the total cost be y yuan and burn x copies.

Computer company: Y 1=8X

School: Y2=4X+ 120

When X=30, Y 1=Y2.

When X & gt30: 00, y1>; Y2

When x

The key to baking

The definition, image and nature of the linear function are the C-level knowledge points in the interpretation of the senior high school entrance examination, especially the D-level knowledge points in the interpretation of the senior high school entrance examination. It is often combined with inverse proportional function, quadratic function and equation, equation and inequality, and appears in the senior high school entrance examination questions in the form of multiple-choice questions, fill-in-the-blank questions and analytical questions, accounting for about 8 points. In order to solve this kind of problems, classification discussion, combination of numbers and shapes, equations and inequalities are often used.

Example 3 If the range of x in the linear function y=kx+b is -2≤x≤6, the range of the corresponding function value is-1 1≤y≤9. Find the analytical expression of this function.

Solution:

(1) If k > 0, the equations can be -2k+b=- 1 1.

6k+b=9

If k=2.5 b=-6, then the functional relationship at this time is y = 2.5x-6.

(2) If k < 0, the equations can be -2k+b=9.

6k+b=- 1 1

If k=-2.5 b=4, then the resolution function at this time is y=-2.5x+4.

The key to baking

This question mainly examines students' understanding of the nature of functions. If K > 0, y will increase with the increase of x; If k < 0, y decreases with the increase of x.

Defining and defining expressions

Generally speaking, there is the following relationship between independent variable x and dependent variable y:

General formula:1:y = ax 2; +bx+c(a≠0, a, b and c are constants), then y is called the quadratic function of x.

Count. Vertex coordinates (-b/2a, (4ac-b 2)/4a)

2. Vertex type: y = a (x-h) 2+k or y = a (x+m) 2+k (the two formulas are essentially the same,

But junior high school textbooks are the first formula)

3. Intersection point (with X axis): y=a(x-x 1)(x-x2)

Important concepts: (a, b, c are constants, a≠0, a determines the opening direction of the function, a >；; 0, the opening direction is upward, a

The right side of a quadratic function expression is usually a quadratic trinomial.

X is an independent variable and y is a quadratic function of X.

X 1, x2 = [-b (b 2-4ac) under the root sign ]/2a (that is, the formula for finding the root of a quadratic equation with one variable).

There are also cross multiplication and collocation to find the root.

[Edit this paragraph] image of quadratic function

Make an image of the square of the quadratic function y=2x in the plane rectangular coordinate system,

It can be seen that the image of quadratic function is an endless parabola. Different quadratic function images

If the drawn figure is accurate, then the quadratic function will be translated by the general formula.

Note: The sketch itself should have an image of 1 with a name function next to it.

Draw the symmetry axis and point out that X=

3 coordinates intersecting with X axis, coordinates intersecting with Y axis, and coordinates of vertices.

[Edit this paragraph] The properties of parabola

1. Parabola is an axisymmetric figure. The symmetry axis is a straight line x = -b/2a.

The only intersection of the symmetry axis and the parabola is the vertex p of the parabola.

Especially when b=0, the symmetry axis of the parabola is the Y axis (that is, the straight line x=0).

2. The parabola has a vertex p, and the coordinate is P (-b/2a, (4ac-b 2)/4a).

-b/2a=0, p is on the y axis; When δ = b 2-4ac = 0, p is on the x axis.

3. Quadratic coefficient A determines the opening direction and size of parabola.

When a > 0, the parabola opens upward; When a < 0, the parabola opens downward.

The larger the |a|, the smaller the opening of the parabola.

4. Both the linear coefficient b and the quadratic coefficient a*** determine the position of the symmetry axis.

When the signs of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis; Because if the axis of symmetry is on the left, the axis of symmetry is less than 0, which is -b/2a.

When the signs of A and B are different (that is, AB < 0), the symmetry axis is on the right side of the Y axis. Because the axis of symmetry is on the right, it is greater than 0, that is,-b/2a >; 0, so b/2a should be less than 0, so a and b should have different signs.

It can be simply recorded as the same as left and right, that is, when the symbols of A and B are the same (that is, AB > 0), the symmetry axis is left on the Y axis; When the symbols of a and b are different.

(i.e. ab < 0), and the symmetry axis is on the right side of the y axis.

In fact, b has its own geometric meaning: the analytical function (linear function) of the tangent line of parabola at the intersection of parabola and Y axis.

The value of the slope k. It can be obtained by taking the derivative of a quadratic function.

5. The constant term c determines the intersection of parabola and Y axis.

The parabola intersects the Y axis at (0, c)

6. Number of intersections between parabola and X axis

When δ = b 2-4ac > 0, the parabola has two intersections with the X axis.

When δ = b 2-4ac = 0, there are 1 intersections between parabola and X axis.

_______

When δ = b 2-4ac < 0, the parabola has no intersection with the X axis. The value of x is the reciprocal of the imaginary number (x =-b √ b 2-4ac, multiplied by.

Imaginary number I, the whole equation divided by 2a)

When a>0, the function obtains the minimum value f (-b/2a) = 4ac-b2/4a at x= -b/2a; At {x | x

{x | x >-b/2a} is an increasing function; The opening of parabola is upward; The range of the function is {y | y ≥ 4ac-b 2/4a}, and vice versa.

When b=0, the axis of symmetry of parabola is the Y axis. At this point, the function is even, and the analytical expression is transformed into y = ax 2+c (a ≠ 0).

7. Special value form

① y=a+b+c when x =1.

② y=a-b+c when x =-1.

③ y=4a+2b+c when ③x = 2.

④ y=4a-2b+c when x =-2.

8. domain: r

Scope: (Corresponding to the analytical formula, and only discussing the case that A is greater than 0, please ask the reader to infer whether A is less than 0) ① [(4ac-b 2)/4a,

Positive infinity); ②[t, positive infinity]

Parity: even function

Periodicity: None

Analytical formula:

①y = ax2+bx+c[ general formula]

⑴a≠0

(2) when a > 0, the parabolic opening is upward; A < 0, parabolic opening downward;

⑶ Extreme point: (-b/2a, (4ac-b2)/4a);

⑷δ=b^2-4ac，

δ> 0, where the image intersects the X axis at two points:

([-b-√δ]/2a, 0) and ([-b+√δ]/2a, 0);

Δ = 0, the image intersects the x axis at one point:

(-b/2a，0)；

δ < 0, the image has no intersection with the X axis;

②y = a(x-h)2+k[ vertex]

At this time, the corresponding extreme point is (h, k), where h=-b/2a and k = (4ac-b2)/4a;

③y=a(x-x 1)(x-x2)[ intersection (dichotomy) ](a≠0)

Axis of symmetry X=(X 1+X2)/2 when a >: 0 and X≥(X 1+X2)/2, y increases with the increase of x, and when a >: 0 and X≤(X 1+X2)/2, y follows x.

Decrease with the increase of …

At this time, x 1 and x2 are the two intersections of the function and the x axis, and the analytical formula can be obtained by substituting x and y (generally connected by a quadratic equation with one variable).

Use).

[Edit this paragraph] Quadratic function and unary quadratic equation

In particular, the quadratic function (hereinafter called function) y = ax 2+bx+c,

When y=0, the quadratic function is a univariate quadratic equation about x (hereinafter referred to as equation).

That is, ax 2+bx+c = 0.

At this point, whether the function image intersects with the X axis means whether the equation has real roots.

The abscissa of the intersection of the function and the x axis is the root of the equation.

1. Quadratic function y = ax 2; ,y=a(x-h)^2； , y = a (x-h) 2+k, y = ax 2+bx+c (among all kinds, a≠0) has the same shape but different positions. Their vertex coordinates and symmetry axes are as follows:

Analytical formula

y=ax^2；

y=ax^2+K

y=a(x-h)^2；

y=a(x-h)^2+k

y=ax^2+bx+c

Vertex coordinates

(0,0)

(0，K)

(h，0)

(h，k)

(-b/2a,4ac-b^2/4a)

axis of symmetry

x=0

x=0

x=h

x=h

x=-b/2a

When h>0, y = a (x-h) 2; The image can be represented by parabola y = ax 2; Move the h unit in parallel to the right,

When h < 0, it is obtained by moving |h| units in parallel to the left.

When h>0, k>0 and parabola y = ax 2; Move H units in parallel to the right, and then move K units upward, and you can get an image of y = a (x-h) 2+k;

When h>0, k<0 and parabola y = ax 2; An image with y = a (x-h) 2-k can be obtained by moving h units in parallel to the right and then moving down | k units;

When h0, move the parabola to the left by |h| units in parallel, and then move it up by K units to get an image with y=a(x+h)2+k;

When h < 0, k<0, move the parabola to the left by |h| units in parallel, and then move it down by |k| units to obtain an image with y=a(x-h)2+k; When translating a parabola up or down, left or right, it can be abbreviated as "up plus down, left plus right minus".

Therefore, the image of parabola Y = AX 2+BX+C (A ≠ 0) is studied, and the general formula is changed to Y = A (X-H) 2 through the formula; In the form of +k, its vertex coordinates, symmetry axis and approximate position of parabola can be clearly determined, which provides convenience for drawing images.

2. the image of parabola y = ax 2+bx+c (a ≠ 0): when a >: 0, the opening is upward, when a.

3. parabola y = ax 2+bx+c (a ≠ 0), if a >；; 0, when x ≤ -b/2a, y decreases with the increase of x; When x ≥ -b/2a, y increases with the increase of x, if a

4. The intersection of the image with parabola y = ax 2+bx+c and the coordinate axis:

(1) The image must intersect with the Y axis, and the coordinate of the intersection point is (0, c);

(2) when △ = b 2-4ac >; 0, the image intersects the x axis at two points A(x? , 0) and B(x? 0), where x 1, x2 is the unary quadratic equation ax 2+bx+c = 0.

(a≠0)。 The distance between these two points AB=|x? -x？ | = ∣△/∣ A ∣ (δ under the root sign of the absolute value of a) In addition, the distance between any pair of symmetrical points on a parabola can be | 2× (-b/2a)-a | (A is the abscissa of one of the points).

When △ = 0, the image has only one intersection with the X axis;

When △0, the image falls above the X axis, and when X is an arbitrary real number, there is y>0; When a<0, the image falls below the X axis, and when X is an arbitrary real number, there is Y.

5. the maximum value of parabola y = ax 2+bx+c: if a>0 (a <; 0), then when x= -b/2a, the minimum (large) value of y = (4ac-b 2)/4a.

The abscissa of the vertex is the value of the independent variable when the maximum value is obtained, and the ordinate of the vertex is the value of the maximum value.

6. Find the analytic expression of quadratic function by undetermined coefficient method.

(1) When the given condition is that the known image passes through three known points or three pairs of corresponding values of known x and y, the analytical formula can be set to the general form:

y=ax^2+bx+c(a≠0).

(2) When the given condition is the known vertex coordinate or symmetry axis or the maximum (minimum) value of the image, the analytical formula can be set as the vertex: y = a (x-h) 2+k (a ≠ 0).

(3) When the given condition is that the coordinates of two intersections between the image and the X axis are known, the analytical formula can be set as two formulas: y=a(x-x? )(x-x？ )(a≠0)。

7. The knowledge of quadratic function can be easily integrated with other knowledge, resulting in more complex synthesis problems. Therefore, the comprehensive question based on quadratic function knowledge is a hot topic in the senior high school entrance examination, which often appears in the form of big questions.